Showing papers on "Square matrix published in 1988"
TL;DR: A 2 x 2 matrix describing an inhomogeneous dielectric thin film is derived from Maxwell's equations, and the inhomogeneity is assumed to vary along the direction perpendicular to the film plane.
Abstract: In optical coating calculations, a 2 × 2 matrix formalism is generally used to describe the effect of each layer on the electromagnetic field. In current applications, the layers are considered homogeneous, and, therefore, the matrix used is restricted to this particular case. In this paper, a 2 × 2 matrix describing an inhomogeneous dielectric thin film is derived from Maxwell’s equations. The inhomogeneity is assumed to vary along the direction perpendicular to the film plane. No restriction is made on the amplitude of its variation. The matrix is illustrated in the case of a rugate filter designed according to Sossi’s Fourier transform technique. An improved approximation for the Fourier transform technique is then introduced.
80 citations
TL;DR: In this article, the authors survey the known forms to which a given complex matrix may be reduced by unitary or general consimilarity and describe a canonical form to which it can be reduced.
74 citations
TL;DR: A triangular subtraction technique with a reduced number of weighting factors is proposed for the calculation of the strip integral and a unified path integral theory is proposed to bridge various models and to clarify the integral interaction when a line or a strip passes through a square pixel matrix.
Abstract: Calculated forward projection techniques have been used for various purposes in computerized tomography applications, and several models have been proposed to simulate the tomography projection process. Since the area weighted strip integral is one of the best models, methods to facilitate the computation of strip integrals would be very useful. In particular, a triangular subtraction technique with a reduced number of weighting factors is proposed for the calculation of the strip integral. A unified path integral theory is also proposed to bridge various models and to clarify the integral interaction when a line or a strip passes through a square pixel matrix. >
49 citations
TL;DR: In this paper, the bearing estimation problem is formulated as a matrix-approximation problem and the columns of a matrix X are formed by the snapshot vectors from an N-element array.
Abstract: The bearing estimation problem is formulated as a matrix-approximation problem. The columns of a matrix X are formed by the snapshot vectors from an N-element array. The matrix X is then approximated by a matrix in the least-square sense. The rank as well as the partial structure of the space spanned by the columns of the approximated X matrix are prespecified. After the approximated X matrix is computed, the bearings of the sources and, consequently, the spatial correlation of the source signals are estimated. The performance of the proposed technique is compared with two existing methods using simulation. The comparison is made in terms of bias, mean-squared error, failure rates, and confidence intervals for the mean and the variance estimates for all three methods at different signal-to-noise ratios. When the sources are moving slowly and the number of snapshot vectors available for processing is large, a simple online adaptive algorithm is suggested. >
47 citations
TL;DR: In this article, moments of arbitrary order as well as a recursive relation are obtained for matrix normally distributed variables, where the moment generating function is differentiated by aid of matrix derivatives, and some more details are given for the first four moments.
Abstract: In order to obtain moments for matrix normally distributed variables the moment generating function is differentiated by aid of matrix derivatives. Moments of arbitrary order as well as a recursive relation are obtained. Further, some more details are given for the first four moments
43 citations
01 Jan 1988
TL;DR: This paper generalize for matrix valued functions a number of well known interpolation problems for scalar rational functions and obtain explicit formulas for the solutions and gives a more systematic and transparent exposition based exclusively on analysis in finite dimensional spaces.
Abstract: In this paper we generalize for matrix valued functions a number of well known interpolation problems for scalar rational functions and obtain explicit formulas for the solutions The realization approach toward the study of rational matrix functions from systems theory serves here as the main tool The main results recently appeared in the literature; here we give a more systematic and transparent exposition based exclusively on analysis in finite dimensional spaces
43 citations
01 Mar 1988
TL;DR: In this paper, it was shown that any n by n matrix with determinant 1 whose entries are real or complex continuous functions on a finite dimensional normal topological space can be reduced to a diagonal form by addition operations if and only if the corresponding homotopy class is trivial.
Abstract: It is shown that any n by n matrix with determinant 1 whose entries are real or complex continuous functions on a finite dimensional normal topological space can be reduced to a diagonal form by addition operations if and only if the corresponding homotopy class is trivial, provided that n 5$ 2 for real-valued functions; moreover, if this is the case, the number of operations can be bounded by a constant depending only on n and the dimension of the space. For real functions and n = 2, we describe all spaces such that every invertible matrix with trivial homotopy class can be reduced to a diagonal form by addition operations as well as all spaces such that the number of operations is bounded. Introduction. Let X be a topological space Rx the ring of all continuous functions X -* R (the reals), Rx the subring of bounded functions. For any natural number n and a ring A, MnA denotes the ring of all n by n matrices over A. A matrix a in MnRX can be regarded as a real matrix depending continuously on a parameter which ranges over X, or as a continuous map X -+ MnR. Assume now that det(a) = 1, i.e. a E SLnRX. We want to reduce a to the identity matrix ln by addition operations, i.e. represent a as a product of elementary matrices ai7, where a E A = RX, 1 < i :$ j < n. Since the subgroup EnA of SLnA generated by all elementary matrices is normal [6], it does not matter whether we use row or column addition operations, or both. Note that, by the Whitehead lemma, every diagonal matrix in SLnA is a product of 4(n -1) elementary matrices (for any commutative ring A), so a matrix a in SLnA, can be reduced to ln if and only if it can be reduced to a diagonal form. When X is a point, so A = Rx = R, it is well known that this can be done. Moreover [3, Remark 10 with sr(R) = m = 1], this can be done using at most (n 1)(3n/2 + 1) addition operations. For an arbitrary X, a homotopy obstruction may exist which prevents the reduction. Namely, the addition operations do not change the homotopy class ir(a) of the corresponding map X -+ SLnR. So if this class is not trivial, the reduction is impossible. Assume now that the homotopy class r(a) is trivial (for example, this is always the case when X is contractible). Is it possible to reduce a to ln by addition operations, i.e. does a belong to the subgroup EnRX of SLnRX generated by elementary matrices)? If yes, how many operations are needed? Received by the editors March 2, 1987 and, in revised form, June 18, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 18F25. ? 1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page
41 citations
TL;DR: A method is presented to solveAx=b by computing optimum iteration parameters for Richardson's method, which supplements the Manteuffel algorithm, developed for the Chebyshev case.
Abstract: A method is presented to solveAx=b by computing optimum iteration parameters for Richardson's method. It requires some information on the location of the eigenvalues ofA. The algorithm yields parameters well-suited for matrices for which Chebyshev parameters are not appropriate. It therefore supplements the Manteuffel algorithm, developed for the Chebyshev case. Numerical examples are described.
35 citations
TL;DR: Analogs of known existence tests for cyclic difference sets are applied to rule out existence for most nonprime values of n and it is shown how such patterns can be used to provide hopping patterns for a frequency-hopped multiple-access system.
Abstract: The author examines square matrices of size n containing dot patterns satisfying the following two restrictions: (1) each column contain precisely one dot, and (2) if the pattern is moved around over a plane tied by the same pattern, when in all positions except the home position there is at most one overlap in dots. From differing viewpoints, there matrices are the characteristic functions of either a certain class of relative difference sets or else a select subset of bent functions. Also, the existence of such an (n*n) matrix implies the existence of a finite projective plane of order n. A family of constructions for such matrices is available when n is prime. A polynomial equation characterizing such matrices and resembling the Hall polynomial equation of cyclic difference sets is presented. Analogs of known existence tests for cyclic difference sets are then applied to rule out existence for most nonprime values of n. It is shown how such patterns can be used to provide hopping patterns for a frequency-hopped multiple-access system. >
33 citations
TL;DR: In this paper, the authors characterized the complex square matrices which are expressible as the product of finitely many positive semidefinite matrices; a matrix T can be expressed as such if and only if T⩾0; moreover, the number of factors can always be limited to five.
31 citations
TL;DR: In this paper, the problem of finding an invertible matrix S such that S -1 AS is upper triangular and S-1 ZS is lower triangular is considered. And sufficient conditions for such an S to exist are presented, in some cases, the conditions are also necessary.
TL;DR: In this paper, the authors considered singular systems of the form E[xdot] = ǫ(x,u,t) where E is a square matrix and may be singular.
Abstract: Singular systems of the form E[xdot] = ƒ(x,u,t) are considered, where E is a square matrix and may be singular. It is assumed that for any ‘admissible’ initial state x(t 0), any control u(t) ∊ U yields one and only one continuous state x(t), and there is one and only one continuous adjoint state λ(t). The formulae for functional variation are derived; the necessary condition for optimality—the maximum principle—is obtained; the boundary conditions for the adjoint equations of the singular systems are given; and the necessary and sufficient condition for optimality of linear singular systems is derived.
11 Apr 1988
TL;DR: A novel algorithm for sensor array signal processing and spatial source parameter estimation, referred to as the method of simultaneous rotations, is introduced, in the same genre as the recently proposed ESPRIT algorithm, which works with two identical arrays.
Abstract: A novel algorithm for sensor array signal processing and spatial source parameter estimation, referred to as the method of simultaneous rotations, is introduced. The algorithm is in the same genre as the recently proposed ESPRIT algorithm, which works with two identical arrays, an X array and a Y array. A major tenet of the originality of the new approach is that it treats the X and Y data matrices as a rectangular matrix pencil and uses the fact that they have the same row space, the source subspace, as well as the same column space, the signal subspace. A novel technique for reducing the problem to that of solving the eigensystem of a square matrix pencil of dimension equal to the number of sources is developed, based on the solution to the Procrustes problem for optimally performing an invariant subspace rotation. Refinements to the algorithm invoking the total-least-squares concept are also presented for the purpose of further exploiting the similarity in structure between the X and Y data matrices. >
23 Feb 1988
TL;DR: In this paper, a fast algorithm is presented which can be used to compute an approximate inverse of a positive definite matrix if that matrix is specified only on a multiple band and the approximate inverse is the matrix that closely matches the partially specified matrix.
Abstract: A fast algorithm is presented which can be used to compute an approximate inverse of a positive definite matrix if that matrix is specified only on a multiple band. The approximate inverse is the inverse of a matrix that closely matches the partially specified matrix. It has zeros in the positions that correspond to unspecified entries in the partially specified matrix. It is closely related to the so-called maximum-entropy extension of this matrix. The algorithm is very well suited for implementation on an array processor.
TL;DR: The proposed method can be used to determine the corresponding left/right solvent of a matrix polynomial such that both right and left solvents have the same eigenspectra.
Abstract: This paper is concerned with the extension of the power method, used for finding the largest eigenvalue and associated eigenvector of a matrix, to its block from for computing the largest block eigenvalue and associated block eigenvector of a non-symmetric matrix. Based on the developed block power method, several algorithms are developed for solving the complete set of solvents and spectral factors of a matrix polynomial, without prior knowledge of the latent roots of the matrix polynomial. Moreover, when any right/left solvent of a matrix polynomial is given, the proposed method can be used to determine the corresponding left/right solvent such that both right and left solvents have the same eigenspectra. The matrix polynomial of interest must have distinct block solvents and a corresponding non-singular polynomial matrix. The established algorithms can be applied in the analysis and/or design of systems described by high-degree vector differential equations and/or matrix fraction descriptions.
TL;DR: In this paper, the authors consider the question of whether there exists an invertible lower triangular matrix L such that L -1 AL is upper triangular, and if so, what can be said about the order in which the eigenvalues of A may appear on the diagonal of L-1 AL.
TL;DR: It is shown that the limit as x approaches infinity of N(x)x equals 6.
TL;DR: In this paper, Friedland, D. Hershkowitz, and H. Schneider characterized real square matrices, i.e., matrices whose positive powers are irreducible M -matrices, by finding a recursive property allowing one to express such a matrix using MMA matrices of smaller order.
TL;DR: In this paper, the authors derived necessary and sufficient conditions for the existence of minima of a cost function in linear time-invariant systems of the form E dx/dt=Ax + Bu where E is a square matrix that may be singular and x is a column vector called the descriptor vector of the system.
Abstract: Linear time-invariant systems of the form E dx/dt=Ax + Bu are considered, where E is a square matrix that may be singular and x is a column vector called the descriptor vector of the system. It is assumed that for any admissible initial descriptor vector x(0—), any control vector u(t) yields one and only one descriptor vector x(t). The problem is this: find a control vector u(t) that will drive the descriptor vector of the system from a fixed vector x(0—) to a (not necessarily fixed) final descriptor vector x(tf) while, together with some (not a priori fixed) final time tf, minimizing a cost functional J = 1/2 1∫0 ( xT Qx + uT Ru) dt. Using elementary matrix and variational techniques, necessary conditions are derived for the existence of minima of J; the problem of finding sufficient conditions for the existence of minima of J is not considered. The general results are applied to the special case E = diag { In−m, 0}, B = [o, Im]. The problem of choosing the matrix Q of the cost functional is investigated...
01 Jun 1988
TL;DR: This paper is a short survey of methods for computing bounds on solutions of a system of linear equations with square matrix whose coefficients as well as the right-hand side components are given by real intervals.
Abstract: This paper is a short survey of methods for computing bounds on solutions of a system of linear equations with square matrix whose coefficients as well as the right-hand side components are given by real intervals.
TL;DR: In this article, a combinatorial method of enumerating the Kekule structures of polyhex is shown to be equivalent to a matrix method, which leads to more rapid enumerations.
TL;DR: In this article, a matrix expression for the covariance matrix of a general second degree matrix polynomial under normality assumptions is derived by applying methods of matrix differentiation to the cumulant generating function.
TL;DR: In this paper, two different methods are developed for factoring Fn into products of tridiagonal and permutation matrices, one based on matrix identities associated with FFTs and the Rader prime algorithm, and the other based on a numerical technique, called minimal-variable oblique elimination.
TL;DR: In this paper, conditions for the existence of a positive definite diagonal matrix W such that matrix W + AtW is positive definite, and the applications of the latter to the determination of the stability of interval matrices are considered.
Abstract: For matrix A, with off–diagonal elements not necessarily of the same sign, conditions are obtained for the existence of a positive definite diagonal matrix W, such that matrix WA + AtW is positive definite; and the applications of the latter to the determination of the stability of interval matrices are considered.
TL;DR: A unified approach is proposed here to solve three problems simultaneously, namely H ∞ -optimization, Hankel approximation and minimal balanced realization for scalar systems, wherein merely the singular-value decomposition of an easily constructed square matrix is required to establish the final solutions.
Abstract: A unified approach is proposed here to solve three problems simultaneously, namely H ∞ -optimization, Hankel approximation and minimal balanced realization for scalar systems. It is shown that all these problems can be reduced to the same standard form wherein merely the singular-value decomposition (SVD) of an easily constructed square matrix is required to establish the final solutions. The formulation is solely in terms of the coefficients of a transfer function and the resulting algorithm can be performed in a very computationally efficient way.
TL;DR: When the undirected graph of a real square matrix is a tree of forest, the authors established finitely computable tests yielding information about the magnitudes and multiplicities to the eigenvalues of the matrix.
07 Dec 1988
TL;DR: An order-recursive method for the computation of a square-root factor of the pseudoinverse of a covariance matrix is given and applications to statistical estimation problems such as square root information filters and smoothers and order- Recursion system identification problems are discussed.
Abstract: An order-recursive method for the computation of a square-root factor of the pseudoinverse of a covariance matrix is given. In particular, if additional random variables are added, then the factor for the augmented covariance matrix is obtained from the factor of the original matrix with computations basically involving singular value decompositions (SVD) of submatrices of the additional matrix elements. The algorithm can be used to partition the computation of the pseudoinverse on a parallel systolic array of processors when the number of processors is less than N/sup 2/. Applications to statistical estimation problems such as square root information filters and smoothers and order-recursive system identification problems are discussed to motivate the method. >
TL;DR: One of the direct corollaries of Theorem 1 is substantiating the fact that any realization ofr-graphical integer-pair sequence can be obtained from any other one byr-switchings preserving edge degrees.
Abstract: The main result of the paper is Theorem 1. It concerns the sets of integral symmetric matrices with given block partition and prescribed row, column and block sums. It is shown that by interchanges preserving these sums we can pass from any two matrices, one from each set, to the other two ones falling “close” together as much as possible. One of the direct corollaries of Theorem 1 is substantiating the fact that any realization ofr-graphical integer-pair sequence can be obtained from any other one byr-switchings preserving edge degrees. This result is also of interest in connection with the problem of determinings-complete properties. In the special cases Theorem 1 includes a number of well-known results, some of which are presented.
TL;DR: A necessary and sufficient condition for the identity matrix to be the unique Lyapunov scaling factor of a given real symmetric matrix A is given in this paper, where the uniqueness is shown to be equivalent to the uniqueness of identity matrix as a scaling D for which the kernels of A and AD are identical.
Abstract: A necessary and sufficient condition for the identity matrix to be the unique Lyapunov scaling factor of a given real symmetric matrix A is given. This uniqueness is shown to be equivalent to the uniqueness of the identity matrix as a scaling D for which the kernels of A and AD are identical.